cycles in music: spaces, experience and applications in ... · figure 5: tonnetz: hyper-hexatonic...

Cycles in Music: Spaces, Experienceand Applications in Music Theory

and CompositionGuilherme Bertissolo

Universidade Federal da [email protected]

Abstract: This paper focuses the idea of cycle and its approaches in music theory, in order to proposefurther application to music composition. The impulse for reseaching cycle was made possible through aprevious research, in which I proposed a conceptual framework for the relationship between music andmovement in Capoeira. Here, I first discuss the idea of cycle, then its theoretical approaches, and finally,some compositional processes based on cycles.

Keywords: cycle, composition, creative process.

Introduction

This paper focuses the idea of cycle and its approaches in music theory, in order to proposefurther application to music composition. The impulse for researching cycle was madepossible through a previous research, in which I proposed a conceptual framework for the

relationship between music and movement in Capoeira1.In the first section, I discuss the idea of cycle, proposing a definition for the term, in relation to

cognition and our bodily experience. In the second section, I propose a short discussion abouttheoretical approaches to cycle, where music and mathematics are co-implicated. Finally, I discusssome compositional processes based on cycles, in order to exemplify the previous discussion.

I. The notion of cycle: a conceptual territory

Cyclicity is understood here as a property which is somewhat characterized through a cycle2. Theterm cycle holds many meanings. We may consider several approaches to define a cycle.

1In my doctoral research – PhD in Music Composition at Universidade Federal da Bahia (Federal University of Bahia– UFBA) – I focused the complex interaction between music and movement, and specially its uses and functions in thecreation of compositional processes, taking as reference a context in which it is not possible to establish a clear distinctionbetween them: the Brazilian Capoeira [1]. Capoeira is a combination of music, dance and martial arts that has beendeveloped in Brazil by West African slaves and its descendants and has been exported all over the world. The combinationof field work and a critique of existing literature on the interaction between music and movement led to the proposition of aconceptual framework, four concepts which are not mutually exclusive: Cyclicity, Sharpness, Circularity and Surpriseness.

2It is important to mention that this approach to cycle in music is based on a previous – and preliminary – effort ina presentation entitled as “A noção de Ciclo em música: concepções e aplicações composicionais” [2], realized at theANPPOM’s Congress, João Pessoa.

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mailto:[email protected]

Journal MusMat • May 2018 • Vol. II, No. 1

From a cognitive point of view, the cycle is an image schema based on our bodily experience[3,p. 362], therefore, it is a basic and easily recognizable concept, since we experience different cycleseveryday.

Brower defines cycle in relation to the container image schema, emphasizing its time dimension:

The CYCLE schema serves to organize our experience of time and the changes bywhich we measure time [...] The CYCLE schema combines easily with the CONTAINERschema, much of its structure. The circle, being closed, can be conceptualized as acontainer for motion around its perimeter. [...] [T]emporal containers may be of fixedduration, as in the case of conventional cycles such as minutes, hours, and weeks, orthey may be subject to expansion and contraction, as in the case of most bodily cycles.Cycles may also be nested, with larger cycles subsuming smaller ones, producinga temporal hierarchy [...]. We observe [...] opposition in many bodily cycles – thealternation of left and right in walking, in and out in breathing, back and forth inswinging. The alternation may be balanced, so that the halves of the cycle mirror oneanother exactly. Or the alternation may be asymmetrical, as in the case of most bodilyprocesses of tension and relaxation, with the climax sometimes coming well after themidpoint of the cycle. A wave can assume an infinite variety of forms, and in fact itis this kind of flexibility that allows the schema to play such a pervasive part in ourunderstanding of temporal experience [3, p. 329-30]

Notwithstanding, we propose a conceptual approximation, in order to create a territory forcycle based in three notions (Figure ??).

modeling transformation

reiteration

Figure 1: Notion of cycle based on three concepts

Modeling is conceived here as a virtual connection between two events which allows us toidentify resemblances and correlations between them. In this sense, it is an element capable ofpromoting the identification of traces, even changing ones, common features of two subsequentevents spread in time. In other words, it is a sort of design, schema or contour. This modeling isthen continually changing and transforming through time.

Therefore, the definition I would like to propose is: in a cycle, a certain modeling reiteratesitself by transformation (a cycle not necessarily implicates a repetition, even tough the repetitionis the most literal way of reiteration). As a matter of fact, The idea of cycle is one of the mostimportant schemas for time. Cycles organize our sense of time. In music, cycle plays an importantrole: cycles of songs, cycles of fifths, interval cycles, and so on, are obvious examples. Nevertheless,the notions of cycle should neither be considered as rigid models nor as chains of literal repetition.Even when they are not assumed, there is no doubt they are sorts of cycles and even the mostliteral way of representing a cycle.

It is important to mention that there is no linearity or cause/effect relation between the threenotions, they permeate each other. In other words, most of the time it is not possible to precisely

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the beginning and/or the ending, once we are always in the midst of the cycle. Thus, one can onlydelimit the scope of a cycle through the formalizing interference of an observer. The notions oftransformation and reiteration imply temporality, as mentioned by Candace Brower [3]. The cycleis one of the image schemas based on our body experience (p. 328).

Some questions still remain: are there any cycles which don’t reiterate? Are there any cycleswhich don’t transform themselves? Are there any cycles with no modelling (more or less literal)?I propose here that the approach of cycle as modelling, transformation and reiteration allows us tounderstand such nebulous concept.

Concerning the previous research [1], the rhythm, the melodic profiles, the movements and thegame itself are cyclic characteristics of Capoeira Regional. For example, Ginga is a basic Capoeiramovement which plays a role as a basis for all movements3.

II. Notions of cycle in some approaches in music theory

Laske’s Epistemology of Composition [7] takes the creative process in music as a “CompositionalLife Cycle”. In this sense, composition is a cycle per se. For him, the cycle has four interdependentlevels: ideas, materials, implementation, and work. The author discusses the complex networkof the creative process, taking both model-based and rule-based composition as complementaryapproaches.

The notion of cycle plays an important hole in contemporary music theory, especially inneo-riemannian theory. Cohn [4] presents a historical overview of this field of study, taking asreference the first edition of Lewin’s seminal book [8]. Cohn focuses on Lewin’s essay, written in1982 five years earlier than the book), in order to propose a series of perspectives for music theory.

As a starting point, the neo-riemannian theory took as reference models of voice-leading appliedto triads, a process also known as triadic post-tonality [4, 11]. Afterward, this model was alsoapplied to other pitch class sets and collections.

It is important to mention the important role of smooth voice-leading and its geometricvisualization. Voice-leading is related to pitch-spaces, and in some ways it offers us forms tounderstand the relationships between aggregates. Morris, for example, asserts [9, p. 95]: “in recentmusic theory, graphs—nodes connected by lines or arrows – have become important tools formodeling music, musical structures, and compositional systems”.

Siciliano [10, p. 222] proposes the maximally smooth cycle of triads as a toggling, as expressedin Figure 2.

Figure 2: toggling: maximally smooth voice leading [10, p. 222]

Notice that the modeling is the half-step voice leading between two subsequent notes. Thetransformation always results in new perfect triadic forms. Figure 3 shows the toogling in ageometric approach.

Figure 4 shows a cycle of chords which took the pitch class set 026 as a starting point. Notethe smoothness of the voice leading, as the modeling consists of whole steps which transform thechords gradually until it gets back to the original chord.

3For more examples of cycle in capoeira, its significance, and how the cycle in the context suggested me this approach,see the PhD dissertation [1].

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Figure 3: Toggling in a geometric form: maximally smooth voice leading [10, p. 225]

Figure 4: Toggling 026 with a whole step voice leading [10, p. 226]

The tonnetz is a network of triads in a pitch space [4, p. 175], where each triad is related to itshomonym and relative forms (Figure 5). This cycle is modeled by four cycles of half-step (markedby circles) in C, C sharp, D and E, presenting its homonym and relative forms in anticlockwise.The central square indicates the hyper-hexatonic system.

Straus [11] proposes a series of concepts through which we may discuss atonal voice-leading.His approach establishes tools for connecting different pitch class sets. We point out importantnotions for cycle, such as Pitch-class voice, Transformational voice leading, Uniformity, Balance,Offset, Consistency, Span, Fuzzy transpositions and Voice-leading smoothness.

In the third edition of his seminal book “Introduction to Post Tonal Theory”, Straus alsoapproaches interval cycles[12, p. 154], with examples by Bartók, Ives and Varèse. Unfortunately, heonly mentioned simple cycles, with a unique interval. Tymoczko [13, p. 107] also proposes cyclesof intervals and its modes of visualization in a geometry of music. In spite of the simple cycleshe proposes, Tymoczko’s work presents interesting insights for understanding music throughgeometry4.

4Due to scope issues, this paper focuses on cycle instead of geometry, Tymoczko’s approach shall not be discussed here

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Figure 5: Tonnetz: hyper-hexatonic system [4, p. 175]

On the other hand, Gollin [6, p. 143] proposes “compound interval cycles” projected in a pitchspace with two or more intervals. Interesting examples are Bartók’s Study Op. 18, 1 (Figure 6)and Scherzo from Suite Op. 14 (Figure 7), in which the modeling is composed respectively by thealternation of major/minor thirds and major thirds/minor seconds. Notice in Figure 8 how cycle4-5 engenders the octatonic collection (Bartok also extensively uses the 4-5 cycle).

Figure 6: Cycle of major and minor thirds in Bartók’s Study Op. 18, 1 [6, p. 144]

As I shall further discuss, the idea of projection of multi-aggregate cycles in pitch spaces is apowerful tool for generating material, which allows us, on the one hand, to implode the sonorityof a chord by the projection of its constitunt intervals, and, on the other hand, to transform thechord in a pitch space.

In his turn, Morris [9] proposes musical applications of minimal graph cycles:

Graphs may be constructed from other graphs called input relations. Strictly speaking,a relation is a graph of two nodes connected by one or two arrows. However, wewill allow input relations to be more complex, assuming they satisfy some context-

in a more detailed way.

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Figure 7: Cycle of major thirds and minor seconds in Scherzo from Suite Op. 14 [6, p. 145]

Figure 8: Cycle 4-5 and the octatonic collection [6, p. 147]

sensitive definition of simplicity and/or basic importance. So, input relations can bethe graphic representation of ordered or unordered sets, partially ordered sets, cycles,and so forth. Both the input relations and the graphs they construct may or may notbe partitioned into disconnected subgraphs [9, p. 100]

Figure 9, Morris shows the graph cycle through which he composed the flute excerpt presentedin 10. Notice how the cycle starts in 0 (C), and moves step by step, forming a square in the upperleft. Then, he starts the square again and departs to the lower part of the cycle, followed by theright part of the Figure. All the paths which make the composition of the excerpt possible areeasily recognizable through the graph.

Finally, it is important to mention the work by Pedro Augusto Dias [5], a PhD Dissertationon combined concentric cycles in structuration of pitch in Thomas Adès. Dias analyzes severalmodelings of cycles and proposes important insights though visualization tools made possible bythe geometry of music. Even the work is focused on Adès’ processes, Dias presents and discussesimportant issues on cycles in music, with several examples both in analysis and composition.

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Figure 9: Morris graph cycle [9, p. 104]

Figure 10: Morris’ music based on the graph cycle [9, p. 104]

III. Some applications of cycles in composition

In this section I briefly discuss some applications of cycles in composition5. Fumebianas is a seriesof works composed in previous research on the relationship between music and movement inCapoeira [1]. The notion of cycle plays an important role in the process of generation of harmonicmaterial in the series. Interval cycles made possible the creation of harmonic spaces, by theprojection of a sonority extracted from the context (pentatonic collection) in relation to the pitchclass set 5-16 (especially in its form 03467). Figure 11 shows the smoothness between the twoaggregates.

0

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Figure 11: Smoothness between pentatonic collection and the pitch class set 5-16

The dialog between the two sonorities allowed me to propose cycles of transpositions of5-16 through interpolations of two subsequent pentatonic intervals. Figure 12 shows theseinterpolations and how they were constructed, and Figure 13 shows the application of the cycle in

5The intent of this section is to provide examples of applications of cycle in composition. I hope just to show a fewexamples, in order to illustrates the issues previously discussed in the paper. For more detailed analysis, the scores andrecordings are available in https://guilhermebertissolo.wordpress.com/.

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https://guilhermebertissolo.wordpress.com/


Fumebianas Nº 5.

Figure 12: Cycles of 5-16 transformed by two subsequent pentatonic intervals

Figure 13: Cycles of 5-16 transformed by two subsequent pentatonic intervals - Application in Fumebianas Nº 5

In Figure 14 I show a sequence of chords based on the projection of the sonorities in a pitchspace. It is important to notice the three reiterations of the cycle, each generating different trichords.Figure 15 shows the application of the material in Fumebianas Nº 5.

It is possible to generate, gradually transforming, multiple cycles of chords in this space, whilemaintaining resemblance to the sonority of Capoeira.

In Fumebianas Nº 4, I proposed two different interval projections, from 5-16 and pentatoniccollection, in order to generate pitch spaces. Thus, I constructed pitch spaces based on cyclicprojections of the interval of each set, as shown in Figure 16.

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C#

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Figure 14: Creating pitch spaces

These pitch spaces were the basic map through which the piece moved. For example, I tookthe modeling 3, 2, 2, 5, 5, 5, 6, 4, 4, 5, 5, 4 and 5 and applied to both cycles, generating differentmaterials from the same path in two different spaces (Figure 17). This type of strategy was appliedthroughout the series.

Finally, in Fumebianas Nº 5, I generated material by an algorithmic process made possiblethrough cycles of superposed patterns of eighth notes. Figure 18 shows the modeling, wheredifferent patterns of eighth notes are separated by pauses. These cycles started with five eighthpauses, and then four, and so on, until we get just one eighth pause.

Figures 19 and 20 show the application of the first and last cycle. It is important to notice thegradual intensification process which takes place through the first part of Fumebianas Nº 5. As the

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Figure 15: Creating pitch spaces: application in Fumebianas Nº 5

Figure 16: Two cycles of intervals

texture gets more and more dense with every new cycle.

IV. Final considerations

Cycles plays an important hole both in composition and music theory. The numerous approachesto cycle allow us to better understand music and its creative process. Cycles are important waysof visualization, once they allow us to think in terms of espacialization.

As I argue above, I believe cycles connect our bodily experience to the “allegedly abstract” pitchspaces. Once we think in terms of metaphorical projections, bodily spaces help us to understand

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Figure 17: Same path in two different pitch spaces

Figure 18: Cycles of patterns

Figure 19: Cycles of patterns: five eighth pauses

pitch spaces and even create them. I propose that our experience of cycle is a key to understandingmusic as a complex of experience, cognition, space, geometry and culture.

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Figure 20: Cycles of patterns: one eighth pause

The concepts of cycle I propose here were inspired by Capoeira’s movements. Therefore, cyclehas a meaningful cultural significance in this context, and my point of view was strongly impreg-nated by this significance. The cognitive meaning of cycle as an image schema is fundamental,as expressed by Brower [3, p. 325], since “image schemas that lend coherence to our bodilyexperience are metaphorically reflected in conventional patterns of melody, harmony, phrasestructure, and form”.

Finally, I argue that cycles and their graphs in pitch spaces are so capable of making senseto us, and the geometry of music is so significant, due to the power of metaphorical mappingthrough which the relationship between notes, chords and harmonies take on meaning throughour embodied cognition. Ultimately, the space in music is eminently cultural. In other words, weexperience music spaces as we perceive bodily spaces, with both cognitive and cultural meaning.Thus, it is fundamental to develop tools for understanding music spaces through visualization, inorder to better analyze and create music.

References

[1] Bertissolo, G. 2013. Composição e capoeira: dinâmicas de compor entre música e movimento.Salvador: Programa de Pós-Graduação em Música/UFBA.

[2] Bertissolo, G. 2012. A noção de ciclo em música: concepções e aplicações composicionais. InXXII Congresso da ANPPOM, João Pessoa. Proceedings... , Programa de Pós-Graduação emMúsica – UFPB.

[3] Brower, C. 2000. A cognitive theory of musical meaning. Journal of Music Theory, v.44, n.2, pp.323–379.

[4] Cohn, R. 1998. Introduction to neo-riemannian theory: a survey and a historical perspective.Journal of Music Theory, v.42, n.2, pp. 167–80.

[5] Dias, P.A. 2016. Caminhos Concêntricos - Ciclos Combinados na Estruturação das Alturas em Obrasde Thomas Adès. Salvador: Programa de Pós-Graduação em Música/UFBA.

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[6] Gollin, E. 2007, Fall. Multi-aggregate cycles and multi-aggregate serial techniques in themusic of béla bartók. Music Theory Spectrum, v.29, n.2, pp. 143–176.

[7] Laske, O. E. 1991. Toward an epistemology of composition. Interface — Journal of New MusicResearch, v.20, n. 3–4, pp. 235–269.

[8] Lewin, D. 2007. Generalized Musical Intervals and Transformations. London/New York: OxfordUniversity Press.

[9] Morris, R. 2010. Some musical applications of minimal graph cycles. Theory and Pratice,pp.95-117.

[10] Siciliano, M. 2005. Toggling cycles, hexatonic systems, and some analysis of early atonalmusic. Music Theory Spectrum, v.27, n.2, pp. 221–48.

[11] Straus, J. N. 2003. Uniformity, balance, and smoothness in atonal voice leading. Music TheorySpectrum, v.25, pp. 305–352.

[12] Straus, J. N. 2005. Introduction to Post-Tonal Theory (3rd ed.). Englewood Cliffs: Prentice Hall.

[13] Tymoczko, D. 2011. A Geometry of Music: Harmony and Counterpoint in the Extended CommonPractice. London/New York: Oxford University Press.

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